Set: Mathematical Set
In set6: R6 Mathematical Sets Interface
Description
Details
Active bindings
Methods
See Also
Examples
Description
A general Set object for mathematical sets. This also serves as the parent class to
intervals, tuples, and fuzzy variants.
Details
Mathematical sets can loosely be thought of as a collection of objects of any kind. The Set class
is used for sets of finite elements, for infinite sets use Interval. These can be
expanded for fuzzy logic by using FuzzySets. Elements in a set cannot be duplicated and ordering
of elements does not matter, Tuples can be used if duplicates or ordering are required.
Active bindings
propertiesReturns an object of class Properties, which lists the properties of the Set. Set
properties include:
-
empty - is the Set empty or does it contain elements?
-
singleton - is the Set a singleton? i.e. Does it contain only one element?
-
cardinality - number of elements in the Set
-
countability - One of: countably finite, countably infinite, uncountable
-
closure - One of: closed, open, half-open
traitsList the traits of the Set. Set traits include:
-
crisp - is the Set crisp or fuzzy?
typeReturns the type of the Set. One of: (), (], [), [], {}
maxIf the Set consists of numerics only then returns the maximum element in the Set. For open
or half-open sets, then the maximum is defined by
upper - 1e-15
minIf the Set consists of numerics only then returns the minimum element in the Set. For open
or half-open sets, then the minimum is defined by
lower + 1e-15
upperIf the Set consists of numerics only then returns the upper bound of the Set.
lowerIf the Set consists of numerics only then returns the lower bound of the Set.
classIf all elements in the Set are the same class then returns that class, otherwise "ANY".
elementsIf the Set is finite then returns all elements in the Set as a list, otherwise "NA".
universeReturns the universe of the Set, i.e. the set of values that can be added to the Set.
rangeIf the Set consists of numerics only then returns the range of the Set defined by
upper - lower
lengthIf the Set is finite then returns the number of elements in the Set, otherwise Inf. See
the cardinality property for the type of infinity.
Methods
Public methods
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Method new()
Create a new Set object.
Usage
Set$new(..., universe = Universal$new(), elements = NULL, class = NULL)
Arguments
...ANY Elements can be of any class except list, as long as there is a unique
as.character coercion method available.
universeSet. Universe that the Set lives in, i.e. elements that could be added to
the Set. Default is Universal.
elementslist. Alternative constructor that may be more efficient if passing objects
of multiple classes.
classcharacter. Optional string naming a class that if supplied gives the set the
typed property. All elements will be coerced to this class and therefore there must be
a coercion method to this class available.
Returns
A new Set object.
Method print()
Prints a symbolic representation of the Set.
Usage
Set$print(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Details
The function useUnicode() can be used to determine if unicode should be used when
printing the Set. Internally print first calls strprint to create a printable representation
of the Set.
Method strprint()
Creates a printable representation of the object.
Usage
Set$strprint(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Returns
A character string representing the object.
Method summary()
Summarises the Set.
Usage
Set$summary(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Details
The function useUnicode() can be used to determine if unicode should be used when
printing the Set. Summarised details include the Set class, properties, and traits.
Method contains()
Tests to see if x is contained in the Set.
Usage
Set$contains(x, all = FALSE, bound = NULL)
Arguments
xany. Object or vector of objects to test.
alllogical. If FALSE tests each x separately. Otherwise returns TRUE only if all x pass test.
boundignored, added for consistency.
Details
x can be of any type, including a Set itself. x should be a tuple if
checking to see if it lies within a set of dimension greater than one. To test for multiple x
at the same time, then provide these as a list.
If all = TRUE then returns TRUE if all x are contained in the Set, otherwise
returns a vector of logicals.
Returns
If all is TRUE then returns TRUE if all elements of x are contained in the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
The infix operator %inset% is available to test if x is an element in the Set,
see examples.
Examples
s = Set$new(elements = 1:5)
# Simplest case
s$contains(4)
8 %inset% s
# Test if multiple elements lie in the set
s$contains(4:6, all = FALSE)
s$contains(4:6, all = TRUE)
# Check if a tuple lies in a Set of higher dimension
s2 = s * s
s2$contains(Tuple$new(2,1))
c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2
Method equals()
Tests if two sets are equal.
Usage
Set$equals(x, all = FALSE)
Arguments
xSet or vector of Sets.
alllogical. If FALSE tests each x separately. Otherwise returns TRUE only if all x pass test.
Details
Two sets are equal if they contain the same elements. Infix operators can be used for:
Equal ==
Not equal !=
Returns
If all is TRUE then returns TRUE if all x are equal to the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
Examples
# Equals
Set$new(1,2)$equals(Set$new(5,6))
Set$new(1,2)$equals(Interval$new(1,2))
Set$new(1,2) == Interval$new(1,2, class = "integer")
# Not equal
!Set$new(1,2)$equals(Set$new(1,2))
Set$new(1,2) != Set$new(1,5)
Method isSubset()
Test if one set is a (proper) subset of another
Usage
Set$isSubset(x, proper = FALSE, all = FALSE)
Arguments
xany. Object or vector of objects to test.
properlogical. If TRUE tests for proper subsets.
alllogical. If FALSE tests each x separately. Otherwise returns TRUE only if all x pass test.
Details
If using the method directly, and not via one of the operators then the additional boolean
argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper
subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a
(non-proper) subset if it is fully contained by, or equal to, the other Set.
Infix operators can be used for:
Subset <
Proper Subset <=
Superset >
Proper Superset >=
Returns
If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
Examples
Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE)
Set$new(1,2) < Set$new(1,2,3) # proper subset
c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper
Set$new(1,2,3) <= Set$new(1,2,3) # proper
Method add()
Add elements to a set.
Usage
Set$add(...)
Arguments
...elements to add
Details
$add is a wrapper around the setunion method with setunion(self, Set$new(...)).
Note a key difference is that any elements passed to ... are first converted to a Set, this
important difference is illustrated in the examples by adding an Interval to a Set.
Additionally, $add first coerces ... to $class if self is a typed-set (i.e. $class != "ANY"),
and $add checks if elements in ... live in the universe of self.
Returns
An object inheriting from Set.
Examples
Set$new(1,2)$add(3)$print()
Set$new(1,2,universe = Interval$new(1,3))$add(3)$print()
\dontrun{
# errors as 4 is not in [1,3]
Set$new(1,2,universe = Interval$new(1,3))$add(4)$print()
}
# coerced to complex
Set$new(0+1i, 2i, class = "complex")$add(4)$print()
# setunion vs. add
Set$new(1,2)$add(Interval$new(5,6))$print()
Set$new(1,2) + Interval$new(5,6)
Method remove()
Remove elements from a set.
Usage
Set$remove(...)
Arguments
...elements to remove
Details
$remove is a wrapper around the setcomplement method with
setcomplement(self, Set$new(...)). Note a key difference is that any elements passed to ...
are first converted to a Set, this important difference is illustrated in the examples by
removing an Interval from a Set.
Returns
If the complement cannot be simplified to a Set then a ComplementSet is returned
otherwise an object inheriting from Set is returned.
Examples
Set$new(1,2,3)$remove(1,2)$print()
Set$new(1,Set$new(1),2)$remove(Set$new(1))$print()
Interval$new(1,5)$remove(5)$print()
Interval$new(1,5)$remove(4)$print()
# setcomplement vs. remove
Set$new(1,2,3)$remove(Interval$new(5,7))$print()
Set$new(1,2,3) - Interval$new(5,7)
Method multiplicity()
Returns the number of times an element appears in a set,
Usage
Set$multiplicity(element = NULL)
Arguments
elementelement or list of elements in the set, if NULL returns multiplicity of all elements
Returns
Value, or list of values, in R+.
Examples
Set$new(1, 1, 2)$multiplicity()
Set$new(1, 1, 2)$multiplicity(1)
Set$new(1, 1, 2)$multiplicity(list(1, 2))
Tuple$new(1, 1, 2)$multiplicity(1)
Tuple$new(1, 1, 2)$multiplicity(2)
Method clone()
The objects of this class are cloneable with this method.
Usage
Set$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
Other sets:
ConditionalSet,
FuzzyMultiset,
FuzzySet,
FuzzyTuple,
Interval,
Multiset,
Tuple
Examples
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# Set of integers
Set$new(1:5)
# Set of multiple types
Set$new("a", 5, Set$new(1))
# Each Set has properties and traits
s <- Set$new(1, 2, 3)
s$traits
s$properties
# Elements cannot be duplicated
Set$new(2, 2) == Set$new(2)
# Ordering does not matter
Set$new(1, 2) == Set$new(2, 1)
## ------------------------------------------------
## Method `Set$contains`
## ------------------------------------------------
s = Set$new(elements = 1:5)
# Simplest case
s$contains(4)
8 %inset% s
# Test if multiple elements lie in the set
s$contains(4:6, all = FALSE)
s$contains(4:6, all = TRUE)
# Check if a tuple lies in a Set of higher dimension
s2 = s * s
s2$contains(Tuple$new(2,1))
c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2
## ------------------------------------------------
## Method `Set$equals`
## ------------------------------------------------
# Equals
Set$new(1,2)$equals(Set$new(5,6))
Set$new(1,2)$equals(Interval$new(1,2))
Set$new(1,2) == Interval$new(1,2, class = "integer")
# Not equal
!Set$new(1,2)$equals(Set$new(1,2))
Set$new(1,2) != Set$new(1,5)
## ------------------------------------------------
## Method `Set$isSubset`
## ------------------------------------------------
Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE)
Set$new(1,2) < Set$new(1,2,3) # proper subset
c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper
Set$new(1,2,3) <= Set$new(1,2,3) # proper
## ------------------------------------------------
## Method `Set$add`
## ------------------------------------------------
Set$new(1,2)$add(3)$print()
Set$new(1,2,universe = Interval$new(1,3))$add(3)$print()
## Not run:
# errors as 4 is not in [1,3]
Set$new(1,2,universe = Interval$new(1,3))$add(4)$print()
## End(Not run)
# coerced to complex
Set$new(0+1i, 2i, class = "complex")$add(4)$print()
# setunion vs. add
Set$new(1,2)$add(Interval$new(5,6))$print()
Set$new(1,2) + Interval$new(5,6)
## ------------------------------------------------
## Method `Set$remove`
## ------------------------------------------------
Set$new(1,2,3)$remove(1,2)$print()
Set$new(1,Set$new(1),2)$remove(Set$new(1))$print()
Interval$new(1,5)$remove(5)$print()
Interval$new(1,5)$remove(4)$print()
# setcomplement vs. remove
Set$new(1,2,3)$remove(Interval$new(5,7))$print()
Set$new(1,2,3) - Interval$new(5,7)
## ------------------------------------------------
## Method `Set$multiplicity`
## ------------------------------------------------
Set$new(1, 1, 2)$multiplicity()
Set$new(1, 1, 2)$multiplicity(1)
Set$new(1, 1, 2)$multiplicity(list(1, 2))
Tuple$new(1, 1, 2)$multiplicity(1)
Tuple$new(1, 1, 2)$multiplicity(2)
set6 documentation built on Oct. 18, 2021, 5:06 p.m.
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Description Details Active bindings Methods See Also Examples
Description
A general Set object for mathematical sets. This also serves as the parent class to intervals, tuples, and fuzzy variants.
Details
Mathematical sets can loosely be thought of as a collection of objects of any kind. The Set class is used for sets of finite elements, for infinite sets use Interval. These can be expanded for fuzzy logic by using FuzzySets. Elements in a set cannot be duplicated and ordering of elements does not matter, Tuples can be used if duplicates or ordering are required.
Active bindings
propertiesReturns an object of class
Properties, which lists the properties of the Set. Set properties include:-
empty- is the Set empty or does it contain elements? -
singleton- is the Set a singleton? i.e. Does it contain only one element? -
cardinality- number of elements in the Set -
countability- One of: countably finite, countably infinite, uncountable -
closure- One of: closed, open, half-open
-
traitsList the traits of the Set. Set traits include:
-
crisp- is the Set crisp or fuzzy?
-
typeReturns the type of the Set. One of: (), (], [), [], {}
maxIf the Set consists of numerics only then returns the maximum element in the Set. For open or half-open sets, then the maximum is defined by
upper - 1e-15
minIf the Set consists of numerics only then returns the minimum element in the Set. For open or half-open sets, then the minimum is defined by
lower + 1e-15
upperIf the Set consists of numerics only then returns the upper bound of the Set.
lowerIf the Set consists of numerics only then returns the lower bound of the Set.
classIf all elements in the Set are the same class then returns that class, otherwise "ANY".
elementsIf the Set is finite then returns all elements in the Set as a
list, otherwise "NA".universeReturns the universe of the Set, i.e. the set of values that can be added to the Set.
rangeIf the Set consists of numerics only then returns the range of the Set defined by
upper - lower
lengthIf the Set is finite then returns the number of elements in the Set, otherwise Inf. See the cardinality property for the type of infinity.
Methods
Public methods
Method new()
Create a new Set object.
Usage
Set$new(..., universe = Universal$new(), elements = NULL, class = NULL)
Arguments
...ANYElements can be of any class exceptlist, as long as there is a uniqueas.charactercoercion method available.universeSet. Universe that the Set lives in, i.e. elements that could be added to the Set. Default is Universal.
elementslist. Alternative constructor that may be more efficient if passing objects of multiple classes.
classcharacter. Optional string naming a class that if supplied gives the set the
typedproperty. All elements will be coerced to this class and therefore there must be a coercion method to this class available.
Returns
A new Set object.
Method print()
Prints a symbolic representation of the Set.
Usage
Set$print(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Details
The function useUnicode() can be used to determine if unicode should be used when
printing the Set. Internally print first calls strprint to create a printable representation
of the Set.
Method strprint()
Creates a printable representation of the object.
Usage
Set$strprint(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Returns
A character string representing the object.
Method summary()
Summarises the Set.
Usage
Set$summary(n = 2)
Arguments
nnumeric. Number of elements to display on either side of ellipsis when printing.
Details
The function useUnicode() can be used to determine if unicode should be used when
printing the Set. Summarised details include the Set class, properties, and traits.
Method contains()
Tests to see if x is contained in the Set.
Usage
Set$contains(x, all = FALSE, bound = NULL)
Arguments
xany. Object or vector of objects to test.
alllogical. If
FALSEtests eachxseparately. Otherwise returnsTRUEonly if allxpass test.boundignored, added for consistency.
Details
x can be of any type, including a Set itself. x should be a tuple if
checking to see if it lies within a set of dimension greater than one. To test for multiple x
at the same time, then provide these as a list.
If all = TRUE then returns TRUE if all x are contained in the Set, otherwise
returns a vector of logicals.
Returns
If all is TRUE then returns TRUE if all elements of x are contained in the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
The infix operator %inset% is available to test if x is an element in the Set,
see examples.
Examples
s = Set$new(elements = 1:5) # Simplest case s$contains(4) 8 %inset% s # Test if multiple elements lie in the set s$contains(4:6, all = FALSE) s$contains(4:6, all = TRUE) # Check if a tuple lies in a Set of higher dimension s2 = s * s s2$contains(Tuple$new(2,1)) c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2
Method equals()
Tests if two sets are equal.
Usage
Set$equals(x, all = FALSE)
Arguments
xSet or vector of Sets.
alllogical. If
FALSEtests eachxseparately. Otherwise returnsTRUEonly if allxpass test.
Details
Two sets are equal if they contain the same elements. Infix operators can be used for:
| Equal | == |
| Not equal | != |
Returns
If all is TRUE then returns TRUE if all x are equal to the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
Examples
# Equals Set$new(1,2)$equals(Set$new(5,6)) Set$new(1,2)$equals(Interval$new(1,2)) Set$new(1,2) == Interval$new(1,2, class = "integer") # Not equal !Set$new(1,2)$equals(Set$new(1,2)) Set$new(1,2) != Set$new(1,5)
Method isSubset()
Test if one set is a (proper) subset of another
Usage
Set$isSubset(x, proper = FALSE, all = FALSE)
Arguments
xany. Object or vector of objects to test.
properlogical. If
TRUEtests for proper subsets.alllogical. If
FALSEtests eachxseparately. Otherwise returnsTRUEonly if allxpass test.
Details
If using the method directly, and not via one of the operators then the additional boolean
argument proper can be used to specify testing of subsets or proper subsets. A Set is a proper
subset of another if it is fully contained by the other Set (i.e. not equal to) whereas a Set is a
(non-proper) subset if it is fully contained by, or equal to, the other Set.
Infix operators can be used for:
| Subset | < |
| Proper Subset | <= |
| Superset | > |
| Proper Superset | >=
|
Returns
If all is TRUE then returns TRUE if all x are subsets of the Set, otherwise
FALSE. If all is FALSE then returns a vector of logicals corresponding to each individual
element of x.
Examples
Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE) Set$new(1,2) < Set$new(1,2,3) # proper subset c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper Set$new(1,2,3) <= Set$new(1,2,3) # proper
Method add()
Add elements to a set.
Usage
Set$add(...)
Arguments
...elements to add
Details
$add is a wrapper around the setunion method with setunion(self, Set$new(...)).
Note a key difference is that any elements passed to ... are first converted to a Set, this
important difference is illustrated in the examples by adding an Interval to a Set.
Additionally, $add first coerces ... to $class if self is a typed-set (i.e. $class != "ANY"),
and $add checks if elements in ... live in the universe of self.
Returns
An object inheriting from Set.
Examples
Set$new(1,2)$add(3)$print()
Set$new(1,2,universe = Interval$new(1,3))$add(3)$print()
\dontrun{
# errors as 4 is not in [1,3]
Set$new(1,2,universe = Interval$new(1,3))$add(4)$print()
}
# coerced to complex
Set$new(0+1i, 2i, class = "complex")$add(4)$print()
# setunion vs. add
Set$new(1,2)$add(Interval$new(5,6))$print()
Set$new(1,2) + Interval$new(5,6)
Method remove()
Remove elements from a set.
Usage
Set$remove(...)
Arguments
...elements to remove
Details
$remove is a wrapper around the setcomplement method with
setcomplement(self, Set$new(...)). Note a key difference is that any elements passed to ...
are first converted to a Set, this important difference is illustrated in the examples by
removing an Interval from a Set.
Returns
If the complement cannot be simplified to a Set then a ComplementSet is returned
otherwise an object inheriting from Set is returned.
Examples
Set$new(1,2,3)$remove(1,2)$print() Set$new(1,Set$new(1),2)$remove(Set$new(1))$print() Interval$new(1,5)$remove(5)$print() Interval$new(1,5)$remove(4)$print() # setcomplement vs. remove Set$new(1,2,3)$remove(Interval$new(5,7))$print() Set$new(1,2,3) - Interval$new(5,7)
Method multiplicity()
Returns the number of times an element appears in a set,
Usage
Set$multiplicity(element = NULL)
Arguments
elementelement or list of elements in the
set, ifNULLreturns multiplicity of all elements
Returns
Value, or list of values, in R+.
Examples
Set$new(1, 1, 2)$multiplicity() Set$new(1, 1, 2)$multiplicity(1) Set$new(1, 1, 2)$multiplicity(list(1, 2)) Tuple$new(1, 1, 2)$multiplicity(1) Tuple$new(1, 1, 2)$multiplicity(2)
Method clone()
The objects of this class are cloneable with this method.
Usage
Set$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
Other sets:
ConditionalSet,
FuzzyMultiset,
FuzzySet,
FuzzyTuple,
Interval,
Multiset,
Tuple
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | # Set of integers
Set$new(1:5)
# Set of multiple types
Set$new("a", 5, Set$new(1))
# Each Set has properties and traits
s <- Set$new(1, 2, 3)
s$traits
s$properties
# Elements cannot be duplicated
Set$new(2, 2) == Set$new(2)
# Ordering does not matter
Set$new(1, 2) == Set$new(2, 1)
## ------------------------------------------------
## Method `Set$contains`
## ------------------------------------------------
s = Set$new(elements = 1:5)
# Simplest case
s$contains(4)
8 %inset% s
# Test if multiple elements lie in the set
s$contains(4:6, all = FALSE)
s$contains(4:6, all = TRUE)
# Check if a tuple lies in a Set of higher dimension
s2 = s * s
s2$contains(Tuple$new(2,1))
c(Tuple$new(2,1), Tuple$new(1,7), 2) %inset% s2
## ------------------------------------------------
## Method `Set$equals`
## ------------------------------------------------
# Equals
Set$new(1,2)$equals(Set$new(5,6))
Set$new(1,2)$equals(Interval$new(1,2))
Set$new(1,2) == Interval$new(1,2, class = "integer")
# Not equal
!Set$new(1,2)$equals(Set$new(1,2))
Set$new(1,2) != Set$new(1,5)
## ------------------------------------------------
## Method `Set$isSubset`
## ------------------------------------------------
Set$new(1,2,3)$isSubset(Set$new(1,2), proper = TRUE)
Set$new(1,2) < Set$new(1,2,3) # proper subset
c(Set$new(1,2,3), Set$new(1)) < Set$new(1,2,3) # not proper
Set$new(1,2,3) <= Set$new(1,2,3) # proper
## ------------------------------------------------
## Method `Set$add`
## ------------------------------------------------
Set$new(1,2)$add(3)$print()
Set$new(1,2,universe = Interval$new(1,3))$add(3)$print()
## Not run:
# errors as 4 is not in [1,3]
Set$new(1,2,universe = Interval$new(1,3))$add(4)$print()
## End(Not run)
# coerced to complex
Set$new(0+1i, 2i, class = "complex")$add(4)$print()
# setunion vs. add
Set$new(1,2)$add(Interval$new(5,6))$print()
Set$new(1,2) + Interval$new(5,6)
## ------------------------------------------------
## Method `Set$remove`
## ------------------------------------------------
Set$new(1,2,3)$remove(1,2)$print()
Set$new(1,Set$new(1),2)$remove(Set$new(1))$print()
Interval$new(1,5)$remove(5)$print()
Interval$new(1,5)$remove(4)$print()
# setcomplement vs. remove
Set$new(1,2,3)$remove(Interval$new(5,7))$print()
Set$new(1,2,3) - Interval$new(5,7)
## ------------------------------------------------
## Method `Set$multiplicity`
## ------------------------------------------------
Set$new(1, 1, 2)$multiplicity()
Set$new(1, 1, 2)$multiplicity(1)
Set$new(1, 1, 2)$multiplicity(list(1, 2))
Tuple$new(1, 1, 2)$multiplicity(1)
Tuple$new(1, 1, 2)$multiplicity(2)
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